Let $Omega subset {mathbb R}^N$ ($N geq 3$) be a $C^2$ bounded domain and $delta$ be the distance to $partial Omega$. We study positive solutions of equation (E) $-L_mu u+ g(| abla u|) = 0$ in $Omega$ where $L_mu=Delta + frac{mu}{delta^2} $, $mu in (0,frac{1}{4}]$ and $g$ is a continuous, nondecreasing function on ${mathbb R}_+$. We prove that if $g$ satisfies a singular integral condition then there exists a unique solution of (E) with a prescribed boundary datum $ u$. When $g(t)=t^q$ with $q in (1,2)$, we show that equation (E) admits a critical exponent $q_mu$ (depending only on $N$ and $mu$). In the subcritical case, namely $1<q<q_mu$, we establish some a priori estimates and provide a description of solutions with an isolated singularity on $partial Omega$. In the supercritical case, i.e. $q_muleq q<2$, we demonstrate a removability result in terms of Bessel capacities.
تحميل البحث